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https://mathoverflow.net/questions/173414 | 6 | **Conjecture:**
For every irrational algebraic number $q$ and natural number $b$, the representation of $q$ on base $b$ contains all the digits $[0,\dots,b-1]$.
**Questions:**
1. Has this conjecture been proved, refuted or neither?
2. If proved:
Is there an estimate of the minimum length of $q\_b$ containing al... | https://mathoverflow.net/users/27456 | Conjecture on irrational algebraic numbers | The conjecture has been neither refuted nor proved. The state of the art, as far as I know, is contained in the papers of [Adamczewski and Bugeaud](http://annals.math.princeton.edu/2007/165-2/p04), in which they show that anything with a very low complexity decimal expansion cannot be an algebraic irrational. The compl... | 14 | https://mathoverflow.net/users/11054 | 173433 | 88,479 |
https://mathoverflow.net/questions/173437 | 2 | Let $G= \left\{ \begin{pmatrix} 1&a&c\\0&1&b\\0&0&0 \end{pmatrix} \mid a,b,c\in \mathbb{R} \right\}$ be the Heisenberg group. Is there a compact codimension one submanifold of $G$ which is totally geodesic? (We fix a left invariant metric for the Heisenberg group)
| https://mathoverflow.net/users/36688 | totally geodesic submanifold of Heisenberg group | The question is studied in detail in [Pat Eberlein's 1994 paper.](http://www.ams.org/journals/tran/1994-343-02/S0002-9947-1994-1250818-2/S0002-9947-1994-1250818-2.pdf) I believe the answer to your specific question is **NO.**
| 4 | https://mathoverflow.net/users/11142 | 173441 | 88,482 |
https://mathoverflow.net/questions/173382 | 6 | I would like a result along the following lines to be true, but haven't been able to locate it in the literature; pointers would be welcome.
Let $X\_{t}$ be a finite-state, irreducible, aperiodic Markov chain.
Let $f$ be a bounded functional, which is mean-zero with respect to the stationary distribution of $X$.
Let
... | https://mathoverflow.net/users/17883 | Approximating Markov chains by Brownian motion | It turns out that to find a whole slew of results of this shape I needed to know that
(a) this sort of statement is called "strong invariance principle" more often than "strong approximation";
(b) the result as requested above is the subject of the book by Walter Philipp and William Stout, Almost sure invariance pr... | 5 | https://mathoverflow.net/users/17883 | 173450 | 88,483 |
https://mathoverflow.net/questions/173449 | 2 | Does anybody know where I could find the proof of the Friedlander–Iwaniec theorem. The link that I find when I search for it is <http://www.pnas.org/content/94/4/1054.full.pdf+html>, but this seems more like an outline of the proof than the proof itself. Also, I have been told that the proof is much longer than 4-5 pag... | https://mathoverflow.net/users/40983 | Proof of the Friedlander–Iwaniec theorem | J. Friedlander and H. Iwaniec, The polynomial $X^2+Y^4$ captures its primes. Ann. of Math. (2) 148 (1998), no. 3, 945--1040.
see:
<http://www.ams.org/mathscinet-getitem?mr=1670069>
<http://www.jstor.org/stable/121034?origin=crossref>
| 2 | https://mathoverflow.net/users/630 | 173451 | 88,484 |
https://mathoverflow.net/questions/173461 | 4 | Let $\mathcal{F}$ denote the class of all functions. Let $U,L:\mathcal{F}\rightarrow\mathcal{F}$ denote the mappings where if $f:X\rightarrow Y$, then $U(f):P(X)\rightarrow P(Y),L(f):P(Y)\rightarrow P(X)$ are the mappings where $U(f)(R)=f[R]$ and $L(f)(R)=f^{-1}[R]$ are the image and inverse image functions. Let $\math... | https://mathoverflow.net/users/22277 | Is the monoid of taking iterated images and inverse images freely generated by the image and inverse image operation? | Yes, it is. The idea is that given an element of $\mathcal{M}$, you can detect whether the last step of it was $L$ or $U$, and then undo the steps one by one to recover a unique expression for it.
First, note that $U$ and $L$ are injective: we can recover any function $f$ from either $U(f)$ or $L(f)$. Now let $M\in\m... | 7 | https://mathoverflow.net/users/75 | 173465 | 88,491 |
https://mathoverflow.net/questions/173460 | 4 | Given four real symmetric matrices $A,B \in \mathbb{R}^{n \times n}$ and $C,D \in \mathbb{R}^{m \times m}$, is there an efficient way to compute the determinant:
$\det|A \otimes C + B \otimes D |$
| https://mathoverflow.net/users/54729 | Determinant of sum of Kronecker products | The case $A$ symmetric positive definite and $B$ symmetric and $C$ positive definite and $D$ symmetric is slightly easier.
In that case, first write $P'AP=I$ and $P'BP=\Delta\_1$, and $Q'CQ=I$ and $Q'DQ=\Delta\_2$, where $\Delta\_1$ and $\Delta\_2$ are diagonal matrices.
Now, write
$(P\otimes Q)'(A\otimes C)(P\oti... | 3 | https://mathoverflow.net/users/8430 | 173466 | 88,492 |
https://mathoverflow.net/questions/173377 | 4 | Let $\mathcal{F}$ and $\mathcal{G}$ be semistable sheaves with the same Hilbert polynomial, i.e. $P(\mathcal{F})=P(\mathcal{G})$ (pay attention that these are Hilbert polynomial and not reduced Hilbert polynomial). If $\mathcal{F}$ or $\mathcal{G}$ is stable, than why is any non-zero morphism $f:\,\mathcal{F}\to\mathca... | https://mathoverflow.net/users/54269 | Sheaves with the same Hilbert polynomial | Suppose $f:F \to G$ is injective, and let $C$ be the cokerel, so we have an exact sequence $0 \to F \to G \to C \to 0$. Since the Hilbert polynomial is additive on short exact sequences, and $P(F)=P(G)$, we get that $P(C)$=0. So for all $m>>0$, $h^0(X,C(m))=P(C)(m)=0$. Serre has a theorem that says that for all $m>>0$ ... | 4 | https://mathoverflow.net/users/19943 | 173476 | 88,500 |
https://mathoverflow.net/questions/173474 | 1 | This question is related to an orthant-type simplex in $\mathbb{R}^n$, which can be defined as
$$ S = \{x\in\mathbb{R}\_{+}^n: \sum\_{i=1}^nx\_i \leq 1\}=\overline{\mbox{conv}}(0,e\_1,\ldots,e\_n). $$
For a nonzero vertex $\bar x=e\_i$, we define the normal cone of $S$ at $\bar x$ as
$$ N\_{S}(\bar x)=\{y\in \mathbb{R}... | https://mathoverflow.net/users/39129 | Volume of normal cone of a simplex (at a vertex) | The volume of the cone is approximately $1/n$ for large $n$. Consider $C=N\_S(e\_n)$.
In order for a vector $y$ to lie in $C$, it suffices that $y\cdot (-e\_n)\le 0$
and $y\cdot (e\_i-e\_n)\le 0$ for each $i<n$.
That is, we require $y\_n\ge 0$ and $y\_n\ge y\_i$ for each $i<n$.
In other words, we need $y\_n=\max(y... | 1 | https://mathoverflow.net/users/11054 | 173477 | 88,501 |
https://mathoverflow.net/questions/173457 | 5 | The third in the "Loop groups and twisted K-theory" series by FHT treats compact Lie groups without any connectedness assumptions. I am trying to unwind what Theorem 2 of that paper (available [here](http://math.berkeley.edu/~teleman/math/finalDirac.pdf), page 5) says for a finite group.
Theorem 2: "For regular $\tau... | https://mathoverflow.net/users/54731 | What does Freed-Hopkins-Teleman say about finite groups? | Take a look at the Annals version of FHT III (2011). There, Theorem 2 is more carefully stated only for compact **connected** Lie $G$. I think what you want is Theorem 3 (whose statement and surrounding context are essentially the same as the '08 version of the paper you linked to):
**Theorem 3**. Let $G$ be a compac... | 6 | https://mathoverflow.net/users/6005 | 173487 | 88,508 |
https://mathoverflow.net/questions/167812 | 4 | I call a profinite group $G$ ***Noetherian***, if evrey ascending chain of closed subgroups is eventually stable. A standart argument shows that every closed subgroup of a Noetherian profinite group is finitely generated.
A profinite group $G$ is called ***just-infinite*** if every nontrivial $M \lhd\_c G$ is open.
... | https://mathoverflow.net/users/38889 | Profinite completions | The answer is **yes** in general.
Since $K$ is finitely generated, by the Nikolov-Segal theorem it coincides with its own profinite completion. So you simply may take $R=K$.
Perhaps, you also want $R$ to be finitely generated. In this case the answer is **no**.
If $K$ is a non-soluble p-adic analytic pro-$p$ gro... | 4 | https://mathoverflow.net/users/10482 | 173493 | 88,511 |
https://mathoverflow.net/questions/173233 | 3 | Let $X$ be a locally compact Hausdorff space, such that $C\_0(X)$ is an injective Banach space, *i.e.* a $\mathfrak{P}\_\lambda$ space for some $\lambda\geq 1$.
* Is it true that $X$ is compact?
* If additionally $X$ is a locally compact group, is it true that $X$ is finite?
| https://mathoverflow.net/users/19593 | On injectivity of the Banach space $C_0(X)$ | $C\_0(G)$ is not injective for infinite $G$, because it is not complemented in $L\_\infty(G)$.
See [Lau, Anthony To Ming; Losert, Viktor. Complementation of certain subspaces of $L\_\infty(G)$ of a locally compact group. Pacific Journal of Mathematics 141 (1990), no. 2, 295--310](http://projecteuclid.org/download/pd... | 3 | https://mathoverflow.net/users/19593 | 173512 | 88,516 |
https://mathoverflow.net/questions/173514 | 3 | Let $V$ be an algebraic variety over a field $K$. Is there a constant $d = d(V) \in \mathbb{N}$ such that for any variety $W$ defined over $K$ and isomorphic to $V$ over the algebraic closure of $K$, there is a field extension $F/K$ with $[F:K] \leq d$ such that $V$ and $W$ are isomorphic over $F$?
| https://mathoverflow.net/users/38889 | Minimal fields of isomorphism for varieties | Yes if $K=\mathbb{R}$ for example, but no in general.
Namely this fails for curves of genus $1$, over $\mathbb{Q}$, say. Given an elliptic curve $E$ over $\mathbb{Q}$ and a positive integer $d$, a general $d$-covering of $E$ will contain a divisor of degree $d^2$, but no divisors of smaller degree. In particular, it ... | 6 | https://mathoverflow.net/users/5101 | 173522 | 88,517 |
https://mathoverflow.net/questions/173535 | 2 | This question is a variant of [this](https://mathoverflow.net/questions/173500) one.
Let $f$ be as in the other question, but suppose that we look at the $\ell$-adic representiation attached to $f$:
$$
\rho\_f : G\_{\mathbb Q} \to \operatorname{GL}\_2(E),
$$
where $E$ is a finite extension of $\mathbb Q\_\ell$, with ... | https://mathoverflow.net/users/54752 | About the restriction of a modular representation to a decomposition subgroup II | The following is (very) well known. Let's assume $f$ is a normalised newform of weight $k \geq 2$, nebentype $\chi$ with $p$th Fourier coefficient $a\_p$. Then your representation $\rho\_{f,p}$ (obtained as a piece of the cohomology of the modular curve) is unramified, and the characteristic polynomial of Frobenius is ... | 0 | https://mathoverflow.net/users/8080 | 173536 | 88,519 |
https://mathoverflow.net/questions/173541 | 4 | The [MDRP theorem](http://www.logicmatters.net/resources/pdfs/MRDP.pdf#page=6) – which answers [Hilbert's tenth problem](http://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem) in the negative – says:
>
> There is no algorithm for determining whether an
> arbitrary diophantine equation has a solution.
>
>
>
... | https://mathoverflow.net/users/2672 | Some types of diophantine equations and their decidability | My first remark is that your question really has little to do with diophantine solutions sets, since by the MRDP theorem these are precisely the computably enumerable sets. So all your questions amount to exactly equivalent questions about the c.e. sets. In other words, every Turing machine program $e$ computes a c.e. ... | 7 | https://mathoverflow.net/users/1946 | 173544 | 88,523 |
https://mathoverflow.net/questions/173534 | 4 | Suppose I have a smooth non-proper algebraic variety $X/\mathbb{C}$.
A vector bundle with flat connection (``differential equation'') on $X$ extends, as was noted by Grothendieck, to a coherent crystal on the infinitesimal site of $X$, realising an equivalence between these two types of object.
Deligne's compariso... | https://mathoverflow.net/users/8080 | Regular singularities and the infinitesimal site | The correspondence can be unpacked from Theorem 6.2 in Kato's "Logarithmic structures of Fontaine-Illusie" (pdf scans available by internet search), in particular by identifying $X = Y = D$, and $S = \operatorname{Spec} \mathbb{C}$. As you suggest, using Nagata and Hironaka, we may choose a smooth compactification with... | 4 | https://mathoverflow.net/users/121 | 173552 | 88,527 |
https://mathoverflow.net/questions/173547 | 6 | The classical de Franchis theorem, as generalized by S. Kobayashi and T. Ochiai ("Meromorphic mappings onto compact complex spaces of general type," *Inventiones*, 1975), states that if $X$ is a complex projective varity of general type, then for any fixed complex projective variety $Y$, there are only finitely many su... | https://mathoverflow.net/users/26522 | Generalizations of de Franchis and function field Mordell | One of the major results in this direction, when $Y$ is either a rational or an elliptic curve, is due to Bogomolov, see
F. Bogomolov, *Families of curves on a surface of general type*, Doklady AN SSSR 236 (1977),
no. 5, 1041-1044, in Russian; English translation: Soviet Math. Dokl. 18 (1977), 1294-1277.
The state... | 5 | https://mathoverflow.net/users/7460 | 173570 | 88,535 |
https://mathoverflow.net/questions/173573 | 6 | Let $G$ be a finite group, $H \leq G$ a proper subgroup. It is well known that the union of the conjugates of $H$ does not cover $G$. I would like to know of more precise results (even in special cases) saying that the union of conjugates misses some structure.
For example, if $G$ is a Frobenius group, the complement... | https://mathoverflow.net/users/38889 | Union of conjugates of a subgroup | To avoid examples like subgroups of cyclic groups, we could assume that $H$ is core-free in $G$. Then the question is equivalent to asking about fixed point free elements of transitive permutation groups.
In a transitive $p$-group, a minimal normal subgroup has order $p$ and must be fixed-point-free (since the stabil... | 9 | https://mathoverflow.net/users/35840 | 173575 | 88,536 |
https://mathoverflow.net/questions/173582 | 2 | Let $\Delta ABC$ be a triangle in the plane. Let $P\_{1}, P\_{2}, P\_{3}$ be the intersection points of bisectors, medians and altitudes, respectively. We define the quantity:
\begin{equation}
Q(\Delta ABC)=\frac{\mathcal{A}(\Delta P{1}P\_{2}P\_{3})}{\mathcal{A}(\Delta ABC)}
\end{equation}
where $\mathcal{A}$ is the... | https://mathoverflow.net/users/36688 | A quantity associated to a triangle | I can say something for the triangles with acute angles. But the obtuse angles may be a bit of a problem, since the intersection of the altitudes will be outside of the triangle.
For the triangle with acute angles, the supremum is definitely less than $1$. In fact, it would be true for any $P\_1$ and $P\_3$, so long ... | 3 | https://mathoverflow.net/users/38468 | 173585 | 88,538 |
https://mathoverflow.net/questions/173498 | 7 | Everyone knows that it's difficult to compute a general fundamental domain for an arithmetic group but are there any specific examples where such domains have been calculated? I'm mostly interested in results related to orthogonal groups of lattices of type O(2, n).
I've only really been able to track down two such r... | https://mathoverflow.net/users/34884 | Examples of fundamental domains | Here are some more examples of fundamental domains for arithmetic
groups, although not of the type $O(2,n)$ you are most interested in.
A situation in which quite something is known on fundamental domains
are the Bianchi groups, i.e., the groups
$PSL\_2(\mathcal{O}\_{\mathbb{Q}(\sqrt{-d})})$ for $d$ a positive
squar... | 6 | https://mathoverflow.net/users/50846 | 173590 | 88,540 |
https://mathoverflow.net/questions/96639 | 14 | It is well known that every $\mathbf{\Pi}^1\_1$-set is a union of $\aleph\_1$-many Borel sets. I wonder whether it can be improved under certain reasonable set theory axioms assumption.
For example, assuming $ZFC+CH$, then it is trivially true that every set is a union of $\aleph\_1$-many closed sets. But this seems ... | https://mathoverflow.net/users/14340 | Decomposing $\mathbf{\Pi}^1_1$ sets into closed sets | Every ${\bf\Pi}\_1^1$ set is the union of $\aleph\_1$ Borel sets, so we only have to make sure that every Borel set is the union of $\aleph\_1$ closed sets.
But every Borel set is analytic and thus a continuous image of the Baire space $\omega^\omega$.
Now the Baire space is the union of $\mathfrak d$ (the dominatin... | 11 | https://mathoverflow.net/users/7743 | 173596 | 88,541 |
https://mathoverflow.net/questions/173592 | 2 | I would be grateful if you could help me with the following question:
Let $n\in \mathbb{N}$ and $k\in \{1, \ldots, n+1\}$. For every $i=1, ..., k$ let $f\_i : \partial\_i \Delta^n \rightarrow \partial\_i \Delta^n$ be an orientation preserving diffeomorphism of the i-th face of the standard $n$-simplex $\Delta^n$. Sup... | https://mathoverflow.net/users/43645 | Extending diffeomorphisms of some faces of the standard $n$- simplex to a diffeomorphism of the $n$-simplex | The answer to your specific question is no. There are some simple technical reasons for why there is a trivial "no" answer, but even if you deal with those there are more substantial reasons why the answer is no in general.
The relatively simple issue is that even if $f\_i$ and $f\_j$ agree on their common boundary,... | 4 | https://mathoverflow.net/users/1465 | 173598 | 88,542 |
https://mathoverflow.net/questions/173588 | 15 | Let $S$ denote the space of rapidly decreasing sequences, which means sequences $a=(a\_k)\_{k=1}^\infty$ such that the numbers $p\_d(a)=\sup\{k^d|a\_k| : 1\leq k<\infty\}$ are finite for all $d\in\mathbb{N}$. We give this space the topology generated by the family of seminorms $p\_d$.
Now let $M$ be a compact smooth ... | https://mathoverflow.net/users/10366 | Bases for spaces of smooth functions | The paper
* MR0688001 Reviewed Vogt, Dietmar Sequence space representations of spaces of test functions and distributions. Functional analysis, holomorphy, and approximation theory (Rio de Janeiro, 1979), pp. 405–443, Lecture Notes in Pure and Appl. Math., 83, Dekker, New York, 1983. (Reviewer: M. Valdivia)
shows t... | 16 | https://mathoverflow.net/users/26935 | 173599 | 88,543 |
https://mathoverflow.net/questions/173411 | 1 | (First time asking question on this forum so please kindly let me know if this is out of scope/inappropriate etc.)
I have a problem that leads me to the following quadratic system of PDEs:-
$
c\_1 w\_q (\partial\_tw\_q) + c\_2 w\_q (\partial\_xw\_q) + c\_3 (\partial\_{x}w\_q)^2 +c\_4 w\_q (\partial\_{xx}w\_q) + c\_... | https://mathoverflow.net/users/54711 | Quadratic PDE Systems | Anyway, let's comb it a bit. Since we start at $1$ at $T$, I'll assume that $w\_q>0$ at least for $t$ close to $T$. Once $w\_0$ is fixed everywhere, we have no choice but to solve left to right.
Put $W=w\_q^p$. Then $(W)\_{xx}=pw^{p-2}(ww\_{xx}+(p-1)w\_x^2)$, so we can choose the power to combine two second order ter... | 2 | https://mathoverflow.net/users/1131 | 173602 | 88,544 |
https://mathoverflow.net/questions/3188 | 22 | Let X be a Banach space, and let Y be a proper non-meager linear subspace of X. If Y is not dense in X, then it is easy to see that the closure of Y has empty interior, contradicting Y being non-meager. So Y must be dense. If Y has the Baire property, then it follows from Pettis Lemma that Y is open and hence closed (s... | https://mathoverflow.net/users/1265 | Are proper linear subspaces of Banach spaces always meager? | I am afraid that Konstantin's accepted answer is seriously flawed.
In fact, what seems to be proved in his answer is that $\ker f$ is of second category, whenever $f$ is a discontinuous linear functional on a Banach space $X$. This assertion has been known as [Wilansky-Klee conjecture](http://www.ams.org/journals/bul... | 23 | https://mathoverflow.net/users/54609 | 173606 | 88,546 |
https://mathoverflow.net/questions/173612 | 6 | Let $G$ be a compact Hausdorff group, $H \leq G$ a closed subgroup of infinite index in $G$.
Is it possible that the conjugates of $H$ cover some open neighbourhood of $1$ in $G$ (or the whole of $G$)?
If this is possible, I would like to know whether there are conditions on $G$ rendering this impossible.
| https://mathoverflow.net/users/38889 | Union of conjugates of a closed subgroup of a compact group | The usual example is $G = {\rm SU}\_2({\bf C})$ and $H$ the diagonal subgroup.
Every unitary matrix is diagonalizable, and thus contained in a
conjugate of $H$.
| 11 | https://mathoverflow.net/users/14830 | 173614 | 88,550 |
https://mathoverflow.net/questions/173617 | 2 | I am interested in the sums
$$g(a,k)=\sum\_{n=0}^{p-1}e^{2\pi i a n^k/p}$$
where $p\equiv1\mod k$ is a prime and $a$ is coprime with $p$.
When $k=2$, it is a [classical fact](http://en.wikipedia.org/wiki/Quadratic_Gauss_sum) that $g(a,2)=\chi(a)g(1,2)$ where $\chi$ is the Legendre symbol.
My question is: Is there an ... | https://mathoverflow.net/users/18698 | Extension of a formula for the quadratic Gauss sums | No, the relation is not as simple in this case. For example, if $k = 3$ and $p = 7$, the three different cubic Gauss sums are roots of $y^{3} - 21y - 7$, and the three roots of this polynomial do not have equal absolute values.
In general, the $k$ different Gauss sums $g(a,k)$ (where $a$ runs over representatives of ... | 10 | https://mathoverflow.net/users/48142 | 173618 | 88,552 |
https://mathoverflow.net/questions/173620 | 3 | Do Heegaard Floer homology detect fibred knot in general oriented 3-manifolds other than $S^3$? If the answer is yes could you give a reference.
| https://mathoverflow.net/users/17492 | Do Heegaard Floer homology detect fibred knot in general oriented 3-manifold? | This is known for any knot in a closed, oriented 3-manifold whose complement is irreducible. Ni proved this in [Knot Floer homology detects fibred knots](http://arxiv.org/abs/math/0607156), building on [Ghiggini](http://arxiv.org/abs/math/0603445)'s proof in the genus 1 case; another proof was given by [Juhász](http://... | 5 | https://mathoverflow.net/users/428 | 173624 | 88,554 |
https://mathoverflow.net/questions/173633 | 18 | Assume that $y/ \log x \rightarrow \infty$ and that $y/x \rightarrow 0$. Then, from a conjecture by [Montgomery and Soundararajan](http://arxiv.org/abs/math/0409258), we expect the number of primes in the interval $[x,x+y]$ to be normally distributed with mean $y/\log x$ and standard deviation $\sqrt{y(\log x/y)/(\log ... | https://mathoverflow.net/users/45947 | The conjecture of Montgomery and Soundararajan on primes in short intervals: Empirical inconsistencies? | There are lower order terms in the work of Montgomery and Soundararajan that may account for the discrepancies you're observing. If you look at Theorem 3 of the paper that you linked, you'll find that the standard deviation should really be
$$
\frac{\sqrt{y (\log \frac xy +B)}}{\log x},
$$
where $B=1-\gamma-\log (2\p... | 24 | https://mathoverflow.net/users/38624 | 173637 | 88,559 |
https://mathoverflow.net/questions/173613 | 14 | Recall that the absolute value of a bounded operator $T$ on a Hilbert space $H$ is the unique positive operator $|T|$ such that $$\||T|x\|=\|Tx\|$$ for all $x\in H$. It can be defined using the continuous functional calculus, or if you have square-roots of positive operators in hand, by $|T|=(T^\*T)^{1/2}$. Likewise, t... | https://mathoverflow.net/users/16107 | How much does the absolute value of an operator behave like an absolute value? | The answer is no. Loewner partial order is subtle. There are some simple examples given in [R. Bhatia, F. Kittaneh, The matrix arithmetic–geometric mean inequality
revisited, Linear Algebra and its Applications 428 (2008) 2177–2191](https://i.stack.imgur.com/NXKCE.png)
| 11 | https://mathoverflow.net/users/54458 | 173638 | 88,560 |
https://mathoverflow.net/questions/173636 | -1 | Suppose on a manifold $M$ we have a differential operator $D$ of order 1 from the smooth sections $C^\infty(M, E)$ to smooth sections $C^\infty(M, F)$, where $E$ and $F$ are vector bundles of rank $n$ and $m$ respectively. If $u \in C^\infty(M, E)$ and $f \in C^\infty(M)$ is a smooth function,
how would $D(fu)$ look l... | https://mathoverflow.net/users/54805 | Generalized Leibniz rule | The "Generalized Leibniz rule" can be written in terms of commutators: see <http://en.wikipedia.org/wiki/Differential_calculus_over_commutative_algebras> and references therein.
In this perspective, the linear function in Stefan Waldmann's comment can be rewritten "without mentioning the $u$" as follows:
$$
\delta\_f... | 1 | https://mathoverflow.net/users/22606 | 173643 | 88,561 |
https://mathoverflow.net/questions/173601 | 9 | Suppose that $P(t)$ is a one-parameter family of rank 2 self-adjoint projections on $\mathbb{C}^n$ that vary analytically in the real parameter $t \in [0,1]$. I claim that there must exist a vector $x \in \mathbb{C}^n$ such that $P(t)x \neq 0$ for all $t$.
In other words, I am hoping to prove that $\bigcup\_{t \in [... | https://mathoverflow.net/users/54756 | A differentiable one-parameter family of codimension 2 subspaces of $\mathbb{C}^n$ cannot fill $\mathbb{C}^n$, right? | Let $Q(t):\mathbb C^n\to\mathbb C^n$ be the orthogonal projection onto $\text{ker}(P(t))$.
Then $t\mapsto Q(t)$ is as differentiable as $P$ was. Now the mapping $(t,x)\mapsto Q(t)(x)$ has rank at most $2n-2+1<2n$. Apply Sard's theorem: The set of regular values is Lebesgue nearly everything. Take a regular value $y$. I... | 5 | https://mathoverflow.net/users/26935 | 173645 | 88,562 |
https://mathoverflow.net/questions/173647 | 4 | Let $p> 3$ be a prime number and $G$ be a finite group of order $p(p^2-1) / 4 $. If $ 4 \mid (p+1)$ then easily we can see that the $ p$-Sylow subgroup of $G$ is a normal subgroup of $G$. As I checked it seems that for each $p$ this result holds i.e. the $ p$-Sylow subgroup of $G$ is a normal subgroup of $G$. Any refer... | https://mathoverflow.net/users/31045 | groups of order $ p(p^2-1) / 4 $ where $p$ is a prime | If the Sylow $p$-subgroup was not normal, then it is not hard to see that the only possibility would be $4|(p-1)$ with $p+1$ Sylow $p$-subgroups. Then $G$ would act $2$-transitively by conjugation on the $p+1$ Sylow $p$-subgroups, with $2$-point stabilizer a factor of $(p-1)/4$.
The finite $2$-transitive permutations... | 7 | https://mathoverflow.net/users/35840 | 173648 | 88,564 |
https://mathoverflow.net/questions/173356 | 26 | Ramanujan mentions in one of his letters to Hardy that $$\frac{1^{5}}{e^{2\pi} - 1}\cdot\frac{1}{2500 + 1^{4}} + \frac{2^{5}}{e^{4\pi} - 1}\cdot\frac{1}{2500 + 2^{4}} + \cdots = \frac{123826979}{6306456} - \frac{25\pi}{4}\coth^{2}(5\pi)$$ If we put $q = e^{-\pi}$ we can see that the series is given by $$\sum\_{n = 1}^{... | https://mathoverflow.net/users/15540 | A sum by Ramanujan for $\coth^{2}(5\pi)$ | Using (see entry 24 on page 291 in Ramanujan's Notebooks II: http://[www.plou](http://www.plouffe.fr/simon/math/Ramanujan%27s%20Notebooks%20II.pdf))
$$\frac{\pi e^{-2\pi z}}{2z[\cosh{(2\pi z)}-\cos{(2\pi z)}]}=
\frac{1}{8\pi z^3}-\frac{1}{4z^2}+\frac{\pi}{4z}-\sum\limits\_{n=1}^\infty
\frac{1}{z^2+(z+n)^2}$$ $$+4z\sum\... | 16 | https://mathoverflow.net/users/32389 | 173651 | 88,565 |
https://mathoverflow.net/questions/173681 | 2 | Suppose $g\_1,...,g\_n\in\mathbb{M}\_{d\times d}(\mathbb{C})$ are matrices and we are interested in finding the smallest matrix Lie algebra containing them, that is, the matrix Lie algebra generated by $g\_1,...,g\_n$.
Specifically, we are interested in obtaining a vector space basis for this algebra.
We are assuming ... | https://mathoverflow.net/users/54824 | Termination conditions for matrix Lie alebra basis generation via "P. Hall algorithm" | Yes. The point of a P. Hall family is just an irredundant way to specify, e.g., a basis of the free Lie algebra on a generating set, avoiding redundancies due to the anticommutativity and Jacobi identity of the Lie bracket. The depth simply specifies how far down the lower central series of the free Lie algebra a brack... | 2 | https://mathoverflow.net/users/38434 | 173684 | 88,574 |
https://mathoverflow.net/questions/173655 | 1 | In section 5.5 of Deligne + Mumford's paper "Irreducibility of the Moduli Space of Curves", they introduce the notion of Teichmuller level structures.
You can find the paper here: <http://publications.ias.edu/sites/default/files/Number7.pdf>
In short, fix a finite group $G$, and let $P$ be the set of primes dividin... | https://mathoverflow.net/users/15242 | how do you define the sheaf of Teichmuller level structures when a section does not exist? | You have a sheaf over $S'$ and, by independence of the choice of section, an isomorphism between the two different pullbacks of the sheaf to $S' \times\_S S'$. This gives descent data to descend the sheaf down to $S$, no?
Maybe a a different characterization would be: The sheafification of the presheaf which associat... | 1 | https://mathoverflow.net/users/18060 | 173687 | 88,576 |
https://mathoverflow.net/questions/173574 | 2 | Let $S$ be a simply connected Riemannan surface . Suppose $\Delta ABC$ is a triangle on $S$. The Area of a triangle is denoted by $\mathcal{A}$. A point $P$ in the interior of $\Delta ABC$ is called "A point of half area" if $\mathcal{A}(\Delta PBC)=\frac{1}{2} \mathcal{A}(\Delta ABC)$. (In planar geometry, $S=\mathbb{... | https://mathoverflow.net/users/36688 | The points of half area of a triangle | I believe that the answer is that the curvature of $S$ has to vanish, i.e., the surface is locally isometric to the plane.
I haven't checked all of the details, which are somewhat messy in my analysis, but here is the basic argument: If the Gauss curvature vanishes identically, then, obviously, the local property hol... | 4 | https://mathoverflow.net/users/13972 | 173689 | 88,577 |
https://mathoverflow.net/questions/173685 | 11 | I saw in Wikipedia the existence of an infinite sequence of Turing degrees $\bf{a\_0}, \bf{a\_1}, \dots$ such that $\bf{a}\_{i+1}' \leq\_T \bf{a}\_i$
where $\bf{a}\_{i+1}'$ is the Turing jump of $\bf{a}\_i$.
However I couldn't find references for this result. Is it a trivial consequence of one of the well-known jump ... | https://mathoverflow.net/users/8833 | Infinite decreasing sequence by the Turing jump | There is surely a proof using only classical recursion theory, but the proof I know uses second-order arithmetic.
Take an $\omega$-model $M$ of $\mathsf{ATR}\_0$ in which there is a countable linear order $L$ such that $M$ satisfies "$L$ is well founded", but such that $L$ is not actually well founded. Then $M$ belie... | 12 | https://mathoverflow.net/users/5442 | 173690 | 88,578 |
https://mathoverflow.net/questions/173688 | 1 | Recall that a function $f: \mathbb{R}^d \longrightarrow \mathbb{C}$ is positive definite, iff for all numbers $N$ and $x\_1, \dots, x\_N \in \mathbb{R}^d$, the matrix $(a\_{ij})$ with entries
$$a\_{ij} = f(x\_j - x\_i)$$
is positive semi-definite.
My question is: Let $g \in L^2(\mathbb{R}^n)$. Is the function
$$f(z, ... | https://mathoverflow.net/users/16702 | Positive Definiteness of a certain function | Define a unitary representation $U$ of $\mathbf R^{2n}$ on $L^2(\mathbf R^n)$ by $(U\_{z,y}g)(w)=g(w+z-y)$. (Special case for $G=\mathbf R^n$ of the so-called two-sided regular representation of $G\times G$ on $L^2(G)$.) Then we have
\begin{align}
(g,U\_{z,y}g)&=\int\_{\mathbf R^n}\overline{g(w)}g(w+z-y)\,dw\\
&=\int\_... | 3 | https://mathoverflow.net/users/19276 | 173695 | 88,579 |
https://mathoverflow.net/questions/121927 | 18 | Let
$$
\Sigma\_g \to E \to \Sigma\_h
$$
be a surface bundle over a surface. Unless otherwise stated, I'll assume $g, h \ge 2$. The theory of Thurston norm shows that surface bundles over $S^1$ often fiber in infinitely many ways (e.g. with fibers of infinitely many genera). For surface bundles over surfaces, however, ... | https://mathoverflow.net/users/960 | Does there exist a surface bundle over a surface of genus at least 2 that fibers in three distinct ways? | Yes. I recently found methods of constructing surface bundles over surfaces with at least $n$ fiberings for any $n$. The idea is to perform a fiberwise connect-sum of trivial bundles in such a way that the two different fiberings for each component can be compatibly assembled into a bundle with many fiberings. See [thi... | 13 | https://mathoverflow.net/users/960 | 173698 | 88,581 |
https://mathoverflow.net/questions/173484 | 8 | A tree $T$ on $\omega$ is *recursively pointed* if it is recursive in each of its branches. We can consider a variant of Sacks forcing where the conditions are recursively pointed perfect trees ordered by inclusion. Given a $V$-generic filter $G$ for this forcing, we can define the real $x\_G = \bigcap\_{T\in G} [T]$. ... | https://mathoverflow.net/users/1682 | Does forcing with recursively pointed perfect trees add a Turing degree that is minimal over $V$? | If you two haven't worked this out already, it seems to me that $x\_G$ is not minimal as a Turing degree above $V$, but is a minimal $V$ degree. For the first, since $x\_G$ computes $0'$, it follows that $x\_G$ is the join of two mutually 1-generic reals, neither of which computes $x\_G$ and at least one of which is no... | 7 | https://mathoverflow.net/users/31026 | 173701 | 88,584 |
https://mathoverflow.net/questions/173674 | 10 | There is a well known theorem which states that:
>
> Theorem(Stallings):
> For any immersion $f$ from a finite graph $D$ to $G$ there is a finite-sheeted covering space $D '$ of $G$ that extends $f$. More precisely, there is an embedding of $D$ into $D'$ and the restriction of the covering map to $D$ coincides wit... | https://mathoverflow.net/users/42001 | Stallings' Theorem for free products of groups | This is one of those 'well known' things that every one does in different ways. I believe the notion of a covering space of a graph of groups was worked out by Bass. The details are rather technical - I prefer to think about coverings of graphs of spaces, which comes to the same thing. (I believe Scott and Wall were th... | 5 | https://mathoverflow.net/users/1463 | 173704 | 88,587 |
https://mathoverflow.net/questions/166136 | 4 | It is well known that given a compact connected smooth manifold without boundary $M$, the set of diffeomorphisms $Diff^{r}(M)$ of $M$ for $r≥1$, is open in $C^{r}(M)$, the set of continuous functions (for example, Thm. 1.7 - Chap 2, in "Differential Topology", M. Hirsch).
My question is: does it follow that $Diff^{r+... | https://mathoverflow.net/users/50774 | The space of diffeomorphisms on a manifold | Yes, $Diff^{r,\alpha}(M)$ is open in $C^{r,\alpha}(M,M)$. You need a Riemannian metric on $M$ to even say what a $C^{r,\alpha}$-map $M\to M$ is. This is because the $C^1$-topology
is coarser than the $C^{r,\alpha}$-topology: so the embedding from $C^{r,\alpha}(M,M)$ with the $C^{r,\alpha}$-topology to $C^{1}(M,M)$ wit... | 4 | https://mathoverflow.net/users/26935 | 173705 | 88,588 |
https://mathoverflow.net/questions/173660 | 5 | Let $\frak{m}$ be the least measurable cardinal. A space $X$ is realcompact if it is homeomorphic to a closed subset of some product $\mathbb{R}^I$. Let $X$ be realcompact with $P\_\frak{m}$ topology, that means - an intersection of fewer than $\frak{m}$ open subsets is again open. A subset $R\subseteq X$ is regular cl... | https://mathoverflow.net/users/16678 | A realcompact analogue of the Baire category theorem | I have a counterexample. In particular, for all regular cardinals $\kappa$ such that there is no uncountable measurable cardinal below $\kappa$, there is a space $X$ that is the union of countably many boundaries of regular open sets but where $X$ is realcompact (and much more).
Let $\kappa$ be an uncountable regular... | 3 | https://mathoverflow.net/users/22277 | 173706 | 88,589 |
https://mathoverflow.net/questions/173694 | 0 | The unboundedness locus L(u) of a plurisubharmonic function u is the set of points x∈X such that u is unbounded in every neighbourhood of x. It always contains the polar locus of u. One knows that the polar locus of a non-trivial psh function is of zero Lebesgue-measure for instance, but is there anything similar known... | https://mathoverflow.net/users/19985 | How large is the unboundedness locus of a plurisubharmonic function? | $L(u)$ can be the whole domain. In dimension $1$, take a dense countable
set $\{ z\_k\}$ and consider the
(pluri) subharmonic function $\sum\_k a\_k\log|z-z\_k|$, where $a\_k>0$ tend to zero sufficiently fast. In other words, a (pluri) polar set where the (pluri)
subharmonic function equals $-\infty$ can be dense.
| 1 | https://mathoverflow.net/users/25510 | 173709 | 88,590 |
https://mathoverflow.net/questions/161673 | 11 | Recently I found [this](http://rads.stackoverflow.com/amzn/click/3110197022) book by Woodin. In the introduction of it the author writes the following:
>
> The main result of this book is the identification of a canonical model in which
> the *Continuum Hypothesis* $(\mathsf{CH})$ is false (1). This model is cano... | https://mathoverflow.net/users/29916 | Canonical model for $\neg\mathsf{CH}$ and $\Omega$-logic | The canonical model that Woodin refers to is the $\mathbb{P}\_{\mathrm{max}}$ extension of $L(\mathbb{R})$, assuming that $L(\mathbb{R})$ satisfies $\mathrm{AD}$ (although the context, the partial order and the model can all be varied to produce equally interesting models). It is the ultimate model for $\mathcal{P}(\om... | 16 | https://mathoverflow.net/users/31807 | 173727 | 88,592 |
https://mathoverflow.net/questions/173707 | 3 | I have a proof for the following assertion which employs Model Theory. It has certainly a pure group theoretic proof, but what is such a proof? Is the assertion trivial?
**Theorem** Let $G$ be a finite group and $s\in G$ be an arbitrary element. Suppose $A=C\_{\mathrm{Aut}(G)}(s)$. Then $C\_G(A)$ is cyclic.
Edition... | https://mathoverflow.net/users/44949 | A double centralizing theorem for finite groups | Here is an expanded version of my comment (it turns out you don't need the generalized Heisenberg group, just the standard one). My claim is that the theorem as stated is false.
Consider the [Heisenberg group](http://en.wikipedia.org/wiki/Heisenberg_group), $H$, defined over $\mathbb{Z}/2$. This finite group may be d... | 10 | https://mathoverflow.net/users/3970 | 174727 | 88,593 |
https://mathoverflow.net/questions/175737 | 10 | Around 1974, Leo Harringto wrote an unpublished note entitled "The constructible reals can be anything", in which he proved that it is consistent that being $\Delta^1\_n$ is the same as being constructible.
Harrington proved his theorem using a version of almost disjoint coding of Jensen and Solovay.
>
> Is ther... | https://mathoverflow.net/users/11115 | Harrington's unpublished note "The constructible reals can be anything" | It seems the Harrington's preprint is here <http://logic-library.berkeley.edu/catalog/detail/2135> but to access it requires CalNet login. Maybe Harrington himself could help if you write to him: <http://math.berkeley.edu/~leo/index.html>
P.S. Prof. Harrington kindly sent me a scan of the preprint. It can be found he... | 10 | https://mathoverflow.net/users/32389 | 175745 | 88,603 |
https://mathoverflow.net/questions/173696 | 2 | Let $F$ be a free profinite group, $G$ a profinite group. Suppose that the free profinite product $F \amalg G$ is a free profinite group. Must $G$ be a free profinite group?
For abstract groups the answer is positive in view of the Nielsen-Schreier theorem.
| https://mathoverflow.net/users/38889 | Making a profinite group free | Following Steinberg comment: If $F$ is free profinite group of infinite rank, and $G$ is projective of rank at most the rank of $F$, then the free product is free. Indeed, it is quasi-free (in the sense that every finite split embedding problem has the rank many solutions) and projective, so free.
**Edit**: Here are ... | 1 | https://mathoverflow.net/users/2042 | 175756 | 88,607 |
https://mathoverflow.net/questions/173604 | 2 | I don't know if this is known, but I was fiddling around with this equality :
$$f:(-1,1)\to (-1,1)\quad \text{satisfies}\quad(f(z)+1)^s=\sum\_{j=0}^{\infty}\dbinom{s}{j}f(z^k)
\quad \forall z\in (-1,1),s\in \mathbb{R}$$
Also you may assume $f$ is differentiable infinitely many times in $(-1,1)$.
Now my questions are :... | https://mathoverflow.net/users/50068 | A functional equality | By MassiveJack's comment it follow that
$$f(z^{\frac{p}{q}})=(f(z))^{\frac{p}{q}}$$
for all $p,q \in \mathbb{N}$ and $0<z<1$ and by continuity
$$f(z^\alpha)=(f(z))^\alpha$$
for all $0<z<1,\alpha>0$.
Now put $z=e^{-1}$ and $\alpha=-\log(x)$ then
$$f(x)=f(e^{-1})^{-\log(x)}=x^{-\log(f(e^{-1}))}$$
hence $f$ is of the form... | 3 | https://mathoverflow.net/users/35593 | 175759 | 88,608 |
https://mathoverflow.net/questions/175740 | 2 | Suppose that I have a surface $S$, smooth proper over an algebraically closed (perfect?)field $k$ that lifts algebraically to some $S\_W$ defined over a field of char 0. I am interested in properties of this lifting.
For exapmle, I read (but without reference, thus I will appreciate some if one knows) that if I have ... | https://mathoverflow.net/users/41314 | Lifting to char 0, references and questions | For an Enriques surface $S$ in odd characteristic, for any lift $S\_W$ over $W$, the restriction map on Picard groups is an isomorphism. Essentially this follows from the fact that $h^2(S,\mathcal{O}\_S)$ equals $0$ together with infinitesimal deformation theory of the Picard functor. I believe this is discussed in Cos... | 3 | https://mathoverflow.net/users/13265 | 175765 | 88,610 |
https://mathoverflow.net/questions/175766 | 1 | Let $A,B$ be two commutative noetherian rings. Let $f:A\to B$ be a formally smooth homomorphism. If $A$ is a regular ring (in the sense that all its localizations are regular local rings), does this imply that $B$ is a regular ring?
| https://mathoverflow.net/users/55859 | Formally smooth map from a regular ring | As I wrote in the comments, the answer is usually yes, depending on the topology you are considering. The two more common cases are (it should be direct references for them, maybe in EGA $0\_{IV}$):
Local case: assume $f$ is local. Let $l$ be the residue field of $B$. You have a Jacobi-Zariski exact sequence in André... | 3 | https://mathoverflow.net/users/36672 | 175772 | 88,613 |
https://mathoverflow.net/questions/175752 | 8 | Let G
be a groupoid. I'm wondering how to construct the free [2-group](http://ncatlab.org/nlab/show/2-group) on G.
By the free 2-group I mean a 2-group $\mathcal{F}\left(G\right)$
equipped with a functor $i:G\longrightarrow\mathcal{F}\left(G\right)$
such that for any 2-group $\mathcal{G}$
and any functor $F:G\lon... | https://mathoverflow.net/users/55851 | How to construct a free 2-group on a groupoid? | let me offer you an alternative. The free 2-group on a groupoid is well defined up to equivalence (not isomorphism), hence I'll offer you a strict model (which is known to exist by abstract reasons, since any 2-group can be strictified).
A strict 2-group is essentially the same thing as a crossed module $\partial\col... | 7 | https://mathoverflow.net/users/12166 | 175777 | 88,614 |
https://mathoverflow.net/questions/175767 | 3 | I have a question related to geometry of numbers, which although seems quite basic, I was rather confused by it so I decided to ask here. Let $\Lambda$ be a lattice in $\mathbb{R}^n$.
Let $R\_1, ..., R\_n$ be the successive minima of $\Lambda$ and $\mathbf{x}^{(1)}, ..., \mathbf{x}^{(n)}$ be the minimal points of $\Lam... | https://mathoverflow.net/users/48408 | Regarding the set up of a geometry of numbers lemma | I think this step is fine: you can rotate the lattice $\Lambda$, or alternately the standard basis in $\mathbb{R}^n$, so that the matrix $[\mathbf{x}^{(1)}, ..., \mathbf{x}^{(n)}]$ becomes upper triangular. The reason is the Iwasawa decomposition which tells us that any invertible matrix is the product of an upper tria... | 6 | https://mathoverflow.net/users/11919 | 175780 | 88,615 |
https://mathoverflow.net/questions/175775 | 29 | To clarify the terms in the question above:
The symmetric group Sym($\Omega$) on a set $\Omega$ consists of all bijections from $\Omega$ to $\Omega$ under composition of functions. A generating set $X \subseteq \Omega$ is minimal if no proper subset of $X$ generates Sym($\Omega$).
This might be a difficult question... | https://mathoverflow.net/users/55865 | Does the symmetric group on an infinite set have a minimal generating set? | I think it follows from Theorem 1.1 of "Subgroups of Infinite Symmetric Groups" by Macpherson and Neumann (J. London Math. Soc. (1990) s2-42 (1): 64-84) that there is no minimal generating set of $S(\Omega)$for infinite $\Omega$.
The theorem states that any chain of proper subgroups of $S(\Omega)$ whose union is $S(\... | 35 | https://mathoverflow.net/users/22989 | 175786 | 88,617 |
https://mathoverflow.net/questions/175778 | 2 | I posted [this question](https://math.stackexchange.com/questions/860973/) on MSE a few time ago, but it did not receive much attention. I thought there might be an elementary answer so didn't want to post it directly on MO. My apologies if this question is not a good fit here.
Let $f:\bar{\mathbb C}\to \bar{\mathbb ... | https://mathoverflow.net/users/24309 | Variation of the argument of a rational function along a circle | There are some easy bounds. Let $f(z) = \dfrac{g\_1(z) g\_2(z)}{h\_1(z) h\_2(z)}$
where $g\_1, g\_2, h\_1, h\_2$ are polynomials not identically $0$, $g\_1$ and $h\_1$ having their roots
inside the circle $C$ and $g\_2$ and $h\_2$ having their roots outside.
Then the length of $g([0,1])$ is at most $2 \pi (\deg(g\_1) +... | 2 | https://mathoverflow.net/users/13650 | 175787 | 88,618 |
https://mathoverflow.net/questions/175761 | 16 | As far as I have heard,Sophus Lie's aim was to construct an analogue of galois theory for differential galois theory. I am familiar with lie group but not with differential galois theory. What is the difference is philosophy between these 2 subjects-Lie groups and Differential Galois Theory?
| https://mathoverflow.net/users/30081 | Differences in philosophy between Lie Groups and Differential Galois Theory | [This is really an extended comment on the remarks of Siegel and Khavkine, but too long for a comment box.] There is an intermediate topic which straddles the world of algebra (where differential Galois theory "really" lives, in the language of differential fields and so on) and analysis (where Lie groups live), namely... | 18 | https://mathoverflow.net/users/52824 | 175788 | 88,619 |
https://mathoverflow.net/questions/175794 | 2 | I would like to find local Darboux coordinates for symplectic structures on coadjoint orbits of some nilpotent Lie group. At first, I thought that this would be not very hard, and that it would be possible to find a change variables by simply playing with the Poisson bracket relations. But eventually I've got stuck.
... | https://mathoverflow.net/users/55876 | How to find Darboux coordinates? | There is a large literature on this. I would recommend the papers
* MR0379752 Vergne, Michèle La structure de Poisson sur l'algèbre symétrique d'une algèbre de Lie nilpotente. Bull. Soc. Math. France 100 (1972), 301–335.
* MR0787118 Kamalin, S. A.; Perelomov, A. M. Construction of canonical coordinates on polarized c... | 5 | https://mathoverflow.net/users/19276 | 175795 | 88,622 |
https://mathoverflow.net/questions/175801 | 5 | It is known in classical probability that if two random variables $X$ and $Y$ obeys
$$\mathbb{E} X^k = \mathbb{E}Y^k, \ \forall \ k \geq 1$$
with additional condition that $\mathbb{E}X^k$ does not grow too fast, then $X$ is equal to $Y$ in distribution. I was wondering that has anybody studied the same problem with the... | https://mathoverflow.net/users/8369 | Approximate Moment Conditions | Here is one approach you can use. The differences $a\_k$ provide you with bounds on the difference between the characteristic functions of the two distributions. This may be easy or not depending on your example. Then you can apply known bounds on the difference between two distributions based on the difference between... | 3 | https://mathoverflow.net/users/9025 | 175814 | 88,627 |
https://mathoverflow.net/questions/52892 | 11 | I have encountered a problem that I suspect has been thoroughly studied but I have not been able to find references. Can anyone point me to a published reference dealing with this or a closely related problem?
Here is the problem:
Let $A$ and $B$ be disjoint $k$-subsets of $\mathbb{Z}/n\mathbb{Z}$. Consider $S(A,B)... | https://mathoverflow.net/users/12419 | Sums of subsets of $\mathbb{Z}/n\mathbb{Z}$ | For convenience I will first consider how many sums $S(A,B)$ are equal
to 0. I might have made some computational errors, but I think that
the method is correct. Regard $n$ as fixed. Let $f\_k$ be the number of
pairs $(A,B)$ of $k$-element subsets (not necessarily disjoint) of
$\mathbb{Z}/n\mathbb{Z}$ such that $\sum\_... | 6 | https://mathoverflow.net/users/2807 | 175815 | 88,628 |
https://mathoverflow.net/questions/175800 | 24 | This question was originally [asked](https://math.stackexchange.com/questions/845018/which-sets-occur-as-boundaries-of-other-sets-in-topological-spaces) on MathStackExchange and is migrated here with [opinion](https://meta.mathoverflow.net/questions/1786/can-i-migrate-this-question-to-mathoverflow) from MO meta. I am i... | https://mathoverflow.net/users/50650 | Which sets occur as boundaries of other sets in topological spaces? | The spaces in which every closed set is a boundary are precisely the resolvable spaces. A topological space is said to be resolvable if it can be partitioned into two dense subspaces.
$\mathbf{Proposition}$ A space is resolvable if and only if every closed set is the boundary of some set.
$\leftarrow$ If $X$ is a t... | 33 | https://mathoverflow.net/users/22277 | 175819 | 88,630 |
https://mathoverflow.net/questions/175807 | 10 | Both Abraham-Marsden and Da Silva seem to imply that given a symplectomorphism $g:T^\ast X\to T^\ast X$ which preserves the tautological $1$-form $\alpha$, it must be that $g$ is fibre preserving.
In fact, this is addressed in the question
[Cotangent bundle lift theorem](https://mathoverflow.net/questions/151083/cot... | https://mathoverflow.net/users/54440 | Lifting a Diffeomorphism to the Cotangent Bundle | The vector field $V$ is the radial vector field in the fibers, and it vanishes along the zero locus of $\alpha$, which is the zero section $Z\subset T^\*M$. Thus, $g$ must preserve $Z$ and hence be a diffeomorphism of $Z$, say $f:Z\to Z$. Since $Z\to M$ is a diffeomorphism, this can be regarded as a diffeomorphism $f:M... | 9 | https://mathoverflow.net/users/13972 | 175824 | 88,633 |
https://mathoverflow.net/questions/175802 | 3 | We know that one of the classical methods for solving some PDEs is the method of separation of variables. It works for known types of PDEs and many examples of physical phenomena are successfully represented in PDE systems where an assumption that the functions are separable in variables seems to work just fine, and we... | https://mathoverflow.net/users/51207 | When is separation of variables an acceptable assumption to solve a PDE? | I think you are asking about the possibility of satisfying the desired boundary conditions by each solution of the Helmholtz equation in the product form $X(x) Y(y)$, where $X(x)$ an $Y(y)$ were obtained by separation of variables. This can only happen if the mode functions $X(x)$ and $Y(y)$ have zero sets that are com... | 3 | https://mathoverflow.net/users/2622 | 175829 | 88,634 |
https://mathoverflow.net/questions/175838 | 8 | This is a continuation of the question [covering properties of strongly compact embedding](https://mathoverflow.net/questions/173263/covering-properties-of-strongly-compact-embedding).
Recall that a cardinal $\kappa$ is $\nu$-strongly compact cardinal if there is an elementary embedding $j:V\rightarrow M$, with criti... | https://mathoverflow.net/users/41953 | Strongly compact cardinal with bad covering properties | The answer is yes, and indeed the situation is not merely
consistent with strong compactness, but rather every strongly
compact cardinal has such embeddings with no small covers.
**Theorem.** Suppose that $\kappa$ is $\delta^+$-strongly compact,
where $\delta^{\lt\kappa}=\delta$. Then there is a
$\delta^+$-strong com... | 3 | https://mathoverflow.net/users/1946 | 175856 | 88,645 |
https://mathoverflow.net/questions/175858 | 2 | A famous result by Kunen regarding elementary embeddings states that there is no such embedding from $V$ onto itself which would be non-trivial. It's clear that if $V=L$ then there is also no non-trivial elementary embedding from $L$ onto itself. By similar means we know that no such embedding, if exists, can be seen f... | https://mathoverflow.net/users/30186 | Kunen's inconsistency concerning $L$ | Your way of describing the [Kunen inconsistency](http://cantorsattic.info/Kunen_inconsistency) is inaccurate in several respects.
* First, the Kunen inconsistency isn't about embeddings of $V$ *onto* $V$, in the sense of surjective maps, since it is trivial to prove in ZF that there are no nontrivial automorphisms o... | 7 | https://mathoverflow.net/users/1946 | 175861 | 88,647 |
https://mathoverflow.net/questions/175860 | 0 | Let $R$ be a complete Henselian discrete valuation ring, $\pi:X \to \mathrm{Spec} (R)$ be a smooth, proper, integral, flat $\mathrm{Spec} (R)$-scheme of dimension $2$. Assume that the genus of the special fiber (which is a smooth curve) is at least $2$. Is there any condition on $X$ such that $H^1(\mathcal{O}\_X)=0$?
... | https://mathoverflow.net/users/46578 | Cohomology of a fibered surface | Let $s,\eta $ be the closed and generic points of $\mathrm{Spec}(R)$. Since you assume $X$ smooth over $R$, we have $h^0(\mathscr{O}\_{X\_\eta })=h^0(\mathscr{O}\_{X\_s})=1$, hence $h^1(\mathscr{O}\_{X\_\eta })=h^1(\mathscr{O}\_{X\_s})\geq 2$. Thus $H^1(X,\mathscr{O}\_X)$ is a $R$-module of rank $\geq 2$, it cannot be ... | 4 | https://mathoverflow.net/users/40297 | 175864 | 88,649 |
https://mathoverflow.net/questions/175854 | 2 | Let $K$ be a field of characteristic zero, and let $\overline{K}$ be its algebraic closure. Let $\overline{M}\_{g,n}(K)$ and $\overline{M}\_{g,n}(\overline{K})$ be the coarse moduli spaces parametrizing Deligne-Mumford stable curves over $K$ and $\overline{K}$ respectively.
Is it true that $\overline{M}\_{g,n}(K)\ti... | https://mathoverflow.net/users/nan | Moduli of curves in characteristic zero | I will just develop a bit the comment by Jason Starr. In these notes
<http://math.stanford.edu/~conrad/papers/coarsespace.pdf>
you can find the following theorem by Keel and Mori:
Let $S$ be a scheme and let $\mathcal{X}$ be an Artin stack that is locally of finite presentation over $S$ and has finite inertia stack $I... | 3 | https://mathoverflow.net/users/14514 | 175867 | 88,650 |
https://mathoverflow.net/questions/175831 | 0 | Let $A$ be a given fixed $n \times m$ matrix. We also consider matrices $B$ of dimension $m \times p$. I am interested in those matrices $A$, for which for all $B \in \mathbb R^{m \times p}$ with non-zero columns, the product $AB$ has at least one row $r\_i$, such that there is **not a single 0** in the row $r\_i$.
I... | https://mathoverflow.net/users/nan | Matrix $A$ such that for all matrices $B$ the product $AB$ has a row with not a single zero | If $\text{Ker}(A) \ne \{0\}$, $A$ certainly doesn't have the property, as you can
take $B$ whose columns are all copies of a nonzero member of $\text{Ker}(A)$ and have
$AB$ all $0$. So assume $\text{Ker}(A) = \{0\}$.
Now the columns of $AB$ are $p$ arbitrary members of $\text{Ran}(A)$ other than $0$. The question is ... | 3 | https://mathoverflow.net/users/13650 | 175868 | 88,651 |
https://mathoverflow.net/questions/175869 | 3 | In "Standard conjectures of algebraic cycles" nLab says:
>
> "... They were also followed by “Beilinson’s dream” on motivic (complexes of) sheaves which comprise so called standard conjectures of Beilinson..."
>
>
>
Did Beilinson publish something of this work?
Which are the publications about this work?
| https://mathoverflow.net/users/nan | Is there any publication of “Beilinson’s dream” on motivic (complexes of) sheaves? | Googling for "Beilinson Conjectures" turns up several good surveys, starting with [this](http://www.math.jussieu.fr/~nekovar/pu/mot.pdf) and [this](https://www.dpmms.cam.ac.uk/~ajs1005/preprints/d-s.pdf). These in turn will point you to the canonical sources: Soule's Bourbaki seminar, Ramakrishnan (Contemporary Mathema... | 1 | https://mathoverflow.net/users/10503 | 175873 | 88,653 |
https://mathoverflow.net/questions/175895 | 4 | The following is a result which I know as a weak form of Jensen's coding lemma$^\*$ (first published in the book "Coding the universe"; also see <http://www.jstor.org/stable/2273986>):
>
> For any class of ordinals $A\subseteq ON$ with $V=L[A]$, there is a class forcing $\mathcal{P}$ such that $\Vdash\_\mathcal{P}V... | https://mathoverflow.net/users/8133 | Question about "Coding the universe" | According to how I understand your question, the answer to your question is no. If $V$ is not of the form $L[x]$, where $x$ is a set, then it is impossible to have a set-forcing extension $V[G]$ of the form $L[a]$, where $a$ is a set. If $V[G]=L[a]$, where $a$ is a set and $G\subset P\in V$ is $V$-generic, then fix a n... | 4 | https://mathoverflow.net/users/1946 | 175901 | 88,666 |
https://mathoverflow.net/questions/175822 | 11 | I am interested in an upper bound on the following incomplete Kloosterman sum
$$ \sum\_{\substack{x=1 \\ x+\_{\_{\bf Z}}x^{-1}>p}}^{p-1}e\left(\frac{x+x^{-1}}{p}\right).$$
Using the Weil's bound it is easy to show that the real part of the sum is bounded by $\sqrt p.$ It is because if $x+x^{-1}>p$ then $(p-x) + (p-x... | https://mathoverflow.net/users/18950 | Incomplete Kloosterman sum | We can estimate this using the Polya-Vinagradov method. We get a main term, which comes from the fact that two elements of $\mathbb F\_p$ that sum to something greater than $p$ are more likely to sum to something a little bit greater than $p$ than a lot, and an error term. The formula is:
$$ \frac{ i p}{2\pi} + O( \s... | 12 | https://mathoverflow.net/users/18060 | 175902 | 88,667 |
https://mathoverflow.net/questions/175892 | 5 | Let $1 \rightarrow A \xrightarrow{a} B \xrightarrow{c} C \rightarrow 1$ be a short exact sequence of topological groups (i.e., all maps are continuous, $A = \mathrm{Ker}(c)$, and $C = \mathrm{Coker}(a)$). Suppose that $A$ and $C$ are compact Hausdorff. Does it follow that $B$ is compact?
| https://mathoverflow.net/users/53197 | Is an extension of compact Hausdorff topological groups compact? | You have to assume that $a$ is a homeomorphism onto its image. Indeed, if you don't then you get counterexamples with $C=1$: let $A$ be a finite group with the discrete topology and let $B$ be the same group with the indiscrete topology. You also have to assume that $c$ is a topological quotient map. Indeed, if you don... | 5 | https://mathoverflow.net/users/52824 | 175906 | 88,670 |
https://mathoverflow.net/questions/175790 | 3 | I'm working with a more or less standard definition of the category **Aut(C)** of automata over a category **C** (where **C** has finite products) which has tuples $$
A=\langle I\_{A},O\_{A},S\_{A},\sigma\_{A}, \omega\_{A},q\_{A}^{0}\rangle$$ as objects, where $I\_{A}$,$O\_{A}$ and $S\_{A}$ are objects in **C**, and $\... | https://mathoverflow.net/users/52812 | Exponential objects in a category of abstract automata | Something doesn't look quite right to me about the claim. Working in $\mathbf{Aut} = \mathbf{Aut}(\mathrm{Set})$, the initial object $\bot$ is the automaton with underlying sets $(I, S, O) = (\emptyset, \{q^0\}, \emptyset)$ with uniquely determined initial state, and empty functions for the transition or next-state fun... | 2 | https://mathoverflow.net/users/2926 | 175913 | 88,673 |
https://mathoverflow.net/questions/175890 | 0 | In *"The Standard Conjectures"* Kleiman says that the following properties of Weil cohomology theory were proved in 1963 for étale cohomology by Artin and Grothendieck, except for the last one that it was proved in 1973 by Deligne.
>
> "... (1) (finiteness), (2) (Poincaré duality), (3) (Künneth formula), (4) (cycle... | https://mathoverflow.net/users/nan | Articles about Weil cohomology theory by Grothendieck and Artin | * Milne, Étale cohomology (book and online version)
* Lei Fu, Étale cohomology theory (essentially an English translation of SGA)
* Freitag-Kiehl, Etale cohomology and the Weil conjecture
| 1 | https://mathoverflow.net/users/nan | 175919 | 88,675 |
https://mathoverflow.net/questions/175911 | 6 | This question has practical meanings in algebraic attack of stream ciphers in cryptography. It can be stated as follows:
Suppose $V$ is a $n$ dimensional vector space over the field $F\_2$, where $F\_2$ is the simplest finite field contains only $\{0,1\}$. you can take $V$ to be $F\_2^n$. $S$ be a subset of $V$. The ... | https://mathoverflow.net/users/55931 | Subset of $F_2^n$ that must contain some subspace of dimension $k$ | As noted by Seva, you're looking for Szemeredi's cube lemma, which is one of the simplest results of its kind.
Let $S\subset\mathbf{F}\_2^n$, and let $\|S\|\_{U^k}^{2^k}$ denote the number of $e\_0,e\_1,\dots,e\_k\in \mathbf{F}\_2^n$ such that
$$\{e\_0 + x\_1 e\_1 + \cdots + x\_k e\_k : x\_i\in\{0,1\}\}\subset S,$$
n... | 8 | https://mathoverflow.net/users/20598 | 175925 | 88,677 |
https://mathoverflow.net/questions/175916 | 20 | A map from an algebraic variety $X$ to a projective space is the same thing as a globally generated line bundle on $X$. What geometric object on $X$ corresponds to a map to a *weighted* projective space?
| https://mathoverflow.net/users/1310 | Maps to projective space == line bundles; what do maps to weighted projective space correspond to? | Let me expand my comment in a short answer.
One of the most common way to build a rational map $f \colon X \dashrightarrow \mathbb{P}(a\_1, \ldots, a\_n)$ is to consider a line bundle $\mathscr{L}$ on $X$ together with sections $$\sigma\_1 \in H^0(X, \, \mathscr{L}^{a\_1}), \quad \sigma\_2 \in H^0(X, \, \mathscr{L}^{... | 20 | https://mathoverflow.net/users/7460 | 175939 | 88,681 |
https://mathoverflow.net/questions/175936 | 6 | Sorry if this sounds like a silly reference request, but I wasn't able to track down any. I'm looking for proof, via forcing, that axiom of choice can fail in a model of $ZF$. All of papers I found where either proving something more sophisticated or, if it was some introductory paper/book, it showed that $\neg CH$ is ... | https://mathoverflow.net/users/30186 | Forcing $\neg AC$ | This is difficult to prove using forcing, for one simple reason.
If $M\models\sf ZFC$, and $G$ is an $M$-generic filter (for some forcing notion), then $M[G]\models\sf ZFC$.
In other words, the only way to use forcing to have a model where the axiom of choice fails, is if the ground model was such that the axiom of... | 11 | https://mathoverflow.net/users/7206 | 175942 | 88,682 |
https://mathoverflow.net/questions/175938 | 3 | The following construction is probably known. I think it should work in any closed symmetric monoidal category, but I will play it safe and formulate the question in the concrete, cartesian closed setting.
Let $\mathcal C$ be a concrete, cartesian closed category. Let $(M,\*,e)$ be a monoid in $\mathcal C$. There is ... | https://mathoverflow.net/users/4814 | Comonads from monoids | This looks like a special case of the fact that given a monad $T$ whose underlying functor has a right adjoint, that right adjoint $C$ acquires a comonad structure (mated to the structure of the monad), and the category of $M$-algebras is equivalent to the category of $C$-coalgebras. (Here $T$ is $M \times -$, with rig... | 8 | https://mathoverflow.net/users/2926 | 175950 | 88,688 |
https://mathoverflow.net/questions/164283 | 6 | While thinking about [this question](https://mathoverflow.net/questions/90046/nilpotent-pro-p-groups), I was led to the following question:
>
> **My question**: Let $G$ be a [topologically finitely generated](http://groupprops.subwiki.org/wiki/Topologically_finitely_generated_group) [pro-$p$](http://en.wikipedia.or... | https://mathoverflow.net/users/3970 | Is any finitely generated nilpotent pro-$p$ group necessarily the pro-$p$ completion of some finitely generated nilpotent group? | Here is an amalgamation of the comments of [Ben Steinberg](https://mathoverflow.net/users/15934/benjamin-steinberg) and [Yves Cornulier](https://mathoverflow.net/users/14094/ycor) with added details. I am grateful to Cornulier for his help over email.
The answer is *no*.
Suppose, for the sake of a contradiction, th... | 6 | https://mathoverflow.net/users/3970 | 175957 | 88,692 |
https://mathoverflow.net/questions/175933 | 1 | Let $Q$ be a non-degenerate indefinite quadratic form on ${\mathbb R}^n$ and write $G=SO(Q)$ for the associated special orthogonal group. Let $K$ be a maximal compact subgroup of $G$ and consider the standard action of $H$ on $V$. It is well-know (and easy to see) that for two vectors $v\_1, v\_2 \in V$ to be on the sa... | https://mathoverflow.net/users/3635 | Orbits of an action of maximal compact subgroups of p-adic orthogonal groups | Let's first review the situation over $\mathbf{R}$, as it gives an excuse to recall Witt's extension theorem that we will use in the non-archimedean case and to see that we had better restrict to dimension at least 3. (The assertion made over $\mathbf{R}$ is false in dimension 2: the locus of nonzero vectors where $Q=0... | 6 | https://mathoverflow.net/users/52824 | 175984 | 88,703 |
https://mathoverflow.net/questions/175926 | 3 | Basically got graph transformation related to graph
isomorphism.
Define $G \to G'$. $V(G')=V(G) \cup E(G)=\{v\_1\ldots v\_n\} \cup \{e\_1\ldots e\_m\}$. Call $v\_i$ vertices $v'$ and $e\_i$ vertices $e'$.
Edges of $G'$.
(1) Add $(v\_i,e\_j)$ iff $v\_i \in e\_j$.
This graph is bipartite and is the subdivision of $... | https://mathoverflow.net/users/12481 | Graph transformation related to graph isomorphism | Here is a proof that the answer to Q1 is **yes**.
Let $f$ be the described transformation and suppose that $G'=f(G)$ for some $G$. Let $A$ and $B$ be the vertices of $G'$ that correspond to the vertices and edges of $G$ respectively. Note that $(A,B)$ is a partition of $V(G')$ such that $A$ is a clique, $B$ is a cli... | 1 | https://mathoverflow.net/users/2233 | 175985 | 88,704 |
https://mathoverflow.net/questions/175978 | 1 | Suppose $G$ is a locally compact group and $H$ is an open subgroup for simplicity. Further suppose $\pi$ is a representation of $H$ on some Hilbert space $\mathcal{H}\_{\pi}$, i.e. $\pi(h)$ is unitary for all $h\in H$, $\pi$ is weakly continuous and $\pi$ is also a homomorphism.
To induce a representation on $G$ from... | https://mathoverflow.net/users/20460 | Is it necessary for $\pi:H\to U(\mathcal{H}_{\pi})$ to be a homomorphism in order for $\text{ind}_H^G\pi$ to be weakly continuous? | I don't believe that weak continuity is *"the only place in which $\pi$ being a homomorphism plays a role"*.
In fact, before you can even talk about weak continuity of the resulting representation, you want $\mathcal H(G, \pi)$ to be a vector subspace of $(\mathcal H\_\pi)^G$ and be $G$-invariant under left translati... | 2 | https://mathoverflow.net/users/19276 | 175990 | 88,706 |
https://mathoverflow.net/questions/175987 | 7 | Please note that this might be some confusion on my part about the work surrounding Vaught's conjecture.
First of all, Vaught Conjecture states that if a first-order complete theory $T$ in a countable language has infinitely many models of size $\aleph\_0$, then either $I(T,\aleph\_0)=\aleph\_0$ or $I(T,\aleph\_0)=2... | https://mathoverflow.net/users/51323 | How do we know if Vaught's Conjecture is Absolute? | The answer is yes. See section 5 of the paper "Bounds on weak scattering" by Sacks (<http://www.math.harvard.edu/~sacks/bws.pdf>); he cites Morley 1970 ("The number of countable models," <http://www.jstor.org/stable/2271150>) as the original proof.
| 5 | https://mathoverflow.net/users/8133 | 175991 | 88,707 |
https://mathoverflow.net/questions/175923 | 26 | The classical version of the van Kampen theorem is concerned about the fundamental group of a based space. In fact, it says that the functor $\pi\_1$ preserves certain types of pushouts in $Top\_\*$.
There is also a generalization of the van Kampen theorem that holds for the fundamental groupoid of a space $X$, where... | https://mathoverflow.net/users/53100 | Why are we interested in the Fundamental Groupoid of a Space? | I confess, in 1965 or so, I first thought that the version of the SvKT (Seifert-van Kampen Theorem) for the fundamental groupoid enabled us to get rid of base points. But then I wanted to calculate the fundamental group of the circle, and gradually realised that we needed $\pi\_1(X,A)$, the fundamental groupoid on a se... | 30 | https://mathoverflow.net/users/19949 | 176007 | 88,714 |
https://mathoverflow.net/questions/176008 | 13 | I am trying to find a good reference for a version of the implicit function theorem over $p$-adic manifolds. None of the texts I have consulted ( including "$p$-adic numbers, $p$-adic analysis, and Zeta functions" by Neal Koblitz and "$p$-adic analysis and Lie groups" by Peter Schneider) seem to discuss it. Of course, ... | https://mathoverflow.net/users/3635 | is there a p-adic implicit function theorem? | There are the lecture notes by Serre (Springer lecture notes 1500) of a course on Lie groups and Lie algebras. He proves the implicit function theorem for analytic functions on a $p$-adic manifold (not smooth functions, though)
The following is a link:
[http://www.amazon.com/Lie-Algebras-Groups-University-Mathematic... | 8 | https://mathoverflow.net/users/23291 | 176011 | 88,715 |
https://mathoverflow.net/questions/175865 | 2 | Sabbah defined a version of b-functions for multiple functions in his 1987 paper here: <http://archive.numdam.org/ARCHIVE/CM/CM_1987__62_3/CM_1987__62_3_283_0/CM_1987__62_3_283_0.pdf>
According to Wikipedia (<http://en.wikipedia.org/wiki/Bernstein%E2%80%93Sato_polynomial>), this b-function factors into linear factors... | https://mathoverflow.net/users/44191 | Sabbah b-functions factoring | I emailed Dr. Sabbah, and he sent me a link to this paper by Gyoja: <http://projecteuclid.org/download/pdf_1/euclid.kjm/1250519266>
That paper proves the theorem on any affine variety.
| 2 | https://mathoverflow.net/users/44191 | 176012 | 88,716 |
https://mathoverflow.net/questions/176006 | 1 | I would like to understand how the Wiener measure of some simple sets can be evaluated.
I will sketch the construction of Wiener measure I have in mind:
We denote the one point compactification of $\mathbb{R}^n$ by $\hat{\mathbb{R}}^n$. Now consider the product space $$\Omega:=\prod\limits\_{0\leq t<\infty}\hat{\mat... | https://mathoverflow.net/users/21870 | How to evaluate the wiener measure of sets? | First let $t$ be fixed and look at the set $\{ w \mid w(t) \in U\}$. Take a sequence of continuous functions $f\_k$ that are uniformly bounded and converge to the indicator function of a set $U$ in the Borel-Algebra of $\mathbb{R}^n$. Let $\pi\_t$ be the projection on the $t$-th component of Omega. Then
$$\int\_\Omega ... | 2 | https://mathoverflow.net/users/16702 | 176017 | 88,717 |
https://mathoverflow.net/questions/175966 | 3 | Let $\pi \_1:SS\to Grpd$ denote the fundamental groupoid functor, from simplicial sets to groupoids, and let $N:Grpd\to SS$ denote the nerve functor. Then $\pi \_1$ is left adjoint to $N.$
On simplicial sets we consider the usual model category and on groupoids the model category structure inherited by the one given ... | https://mathoverflow.net/users/55956 | Does the nerve functor (resp. fundamental groupoid functor) preserve homotopy colimits (resp. homotopy limits)? | Just for fun, here's a purely abstract way of seeing neither of these can happen:
The functors you wrote down both preserve weak equivalences, so are morphisms of relative categories, and induce adjunctions of the associated $\left(\infty,1\right)$-categories, which are $\infty$-groupoids and $1$-groupoids (which is ... | 6 | https://mathoverflow.net/users/4528 | 176027 | 88,721 |
https://mathoverflow.net/questions/176036 | 5 | Let $X$ be a completely regular (Tychonoff) topological space. It is known that if $\mathscr F\subseteq C(X,[0,1])$ separates points and closed sets (that is, for every closed set $E\subseteq X$ and $x\in X\setminus E$, $\exists f\in\mathscr F$ such that $f(x)\notin\operatorname{cl}[f(E)]$), then $X$ can be densely emb... | https://mathoverflow.net/users/55976 | Are Hausdorff compactifications of a Tychonoff space $X$ in one-to-one correspondence with completely regular subalgebras of $BC(X)$? | Here's a counterexample if you don't assume your algebra is self-adjoint. Let $X$ be the unit disk in $\mathbb{C}$ endowed with the discrete topology. Let $B$ be the algebra of complex-valued functions on $X$ that extend continuously to the one-point compactification. Note that every element of $B$ is constant on a coc... | 3 | https://mathoverflow.net/users/75 | 176046 | 88,727 |
https://mathoverflow.net/questions/176054 | 4 | I have a friend who wants to study something applied to neurosciences. He is going to begin his grad studies in mathematics.
He asked me which areas of mathematics could be applied to neurosciences.
Since I don't know the answer, I thought mathoverflow would be the right place to ask.
I mean, there are many areas of m... | https://mathoverflow.net/users/18017 | Which fields could be applied to neurosciences? | My wife is a neuroscientist. I can tell you what she uses:
$\bullet$ a LOT of statistics.
$\bullet$ signal processing (such as wavelet transform).
$\bullet$ some baysian probability theory.
| 11 | https://mathoverflow.net/users/5690 | 176063 | 88,735 |
https://mathoverflow.net/questions/175979 | 8 | If $f$ and $g$ are two functions, define $f \sim g$ if they differ only finitely often on their common domain.
The following property of a large cardinal arose from a problem in model theory. I am interested in its strength. Say that a cardinal $\kappa$ has the weak tree property if the following holds:
Suppose $(... | https://mathoverflow.net/users/26705 | A Weakening of the Tree Property | I give here some upper bounds for the consistency of the weak tree property. Namely - for every successor of regular (including double successors of singulars) we can get the weak tree property by collapsing a weakly compact, and we can get the weak tree property everywhere by collapsing a strongly compact cardinal to ... | 4 | https://mathoverflow.net/users/41953 | 176066 | 88,737 |
https://mathoverflow.net/questions/176072 | 6 | A *partial commutative monoid* (PCM) is, roughly speaking, a set with a partially defined binary operation that is as associative as it can be (given that not all products are defined) and commutative. These things seem to come up in quantum logic, for example:
D. J. Foulis, M. K. Bennet, Effect algebras and unsharp ... | https://mathoverflow.net/users/4910 | Does the category PCM (partial commutative monoids) have a closed symmetric monoidal product? | Yes, it does; see Theorem 11 of [this paper](http://dx.doi.org/10.1007/s10701-012-9654-8).
| 5 | https://mathoverflow.net/users/10368 | 176077 | 88,741 |
https://mathoverflow.net/questions/176070 | 2 | Let M be a compact manifold. The diagonal $M \rightarrow M \times M$ induces
co-product on singular cohomolgy $H^\*(M) \rightarrow H^\*(M) \times H^\*(M)$ via Poincare duality.
I would like to know if there is a co-product on minimal Sullivan model $(\Lambda V, d)$ for M which induces the above co-product, at least... | https://mathoverflow.net/users/27330 | natural co-product on minimal Sullivan model | The short answer is: yes (in the very special case you mention).
Here is a longer answer. Note that the map you are talking about is not really a coproduct on $H^\ast(M)$, since it raises degree by $m$, the dimension of $M$. In fact you have a map
$$
\triangle\_!:H^\ast(M)\to H^{\ast+m}(M\times M).
$$ This map is giv... | 2 | https://mathoverflow.net/users/8103 | 176080 | 88,742 |
https://mathoverflow.net/questions/175983 | 0 | Let $\phi : \mathbb{Q}\_{>0} \to \mathbb{Z}$ be the group morphism defined by $\phi(p) = p$ for $p$ a prime number.
It follows that $\phi(\prod\_i p\_i^{n\_i}) = \sum\_i n\_i p\_i$, with $p\_i$ a prime number and $n\_i \in \mathbb{Z}$.
Let $v: \mathbb{Q}\_{>0} \to \mathbb{N}$ be the map defined by $v(\prod\_i p\... | https://mathoverflow.net/users/34538 | Goldbach-type problem: the valuation of irreducible elements of a subgroup of $\mathbb{Q}_{> 0}$ | **Yes**, here is a proof by induction on an integer $n$, assuming the [**Goldbach conjecture**](http://en.wikipedia.org/wiki/Goldbach%27s_conjecture), or alternatively, just using the *[Goldbach's weak conjecture](http://en.wikipedia.org/wiki/Goldbach%27s_weak_conjecture)* (see at the end), *proved* in 2013, by [Harald... | 0 | https://mathoverflow.net/users/34538 | 176085 | 88,745 |
https://mathoverflow.net/questions/176094 | 2 | **EDIT:** Let $f\colon X\to C$ be a flat proper morphism of complex algebraic (or analytic) varieties. Assume the special fiber over a point $p\in C$ is smooth.
**Is it true that there exists a neighborhood $U$ of $p$ (in Zariski or analytic topology) such that all fibers of $f$ over $U$ are smooth? Or more precisely... | https://mathoverflow.net/users/16183 | When flatness of a morphism implies smoothness? | No if you don't assume that $f$ is proper. For a typical example, let $S$ be a curve with one singular point $s$; take $X=(S\times C) \smallsetminus \{(s,p)\} $.
On the other hand if you assume $f$ proper, this is (trivially) true: the locus of points $x\in X$ such that $T\_x(f)$ is not surjective is a closed subset ... | 7 | https://mathoverflow.net/users/40297 | 176097 | 88,752 |
https://mathoverflow.net/questions/176035 | 2 | Let $T$ denote an algebraic theory.
>
> **Terminological Question.** Let $X$ denote a $T$-algebra. Is there a name for the preorder $\mid$ defined on $X$ by asserting that $a \mid b$ iff there is a term operation $f : X^n \times X \rightarrow X$ such that $f(\tilde{x},a)=b$ for some $\tilde{x} \in X^n$?
>
>
>
... | https://mathoverflow.net/users/26080 | Which algebraic theories have the property that $\mid$ is antisymmetric for all free algebras? | Here is an idea to construct examples generalizing those above. It certainly works for varieties of algebras with a binary operation and should work more generally.
I prefer to give an answer using the old-fashioned language of universal algebra. Suppose we have some signature $\Omega$. Infinitely many operations ar... | 4 | https://mathoverflow.net/users/15934 | 176101 | 88,754 |
https://mathoverflow.net/questions/176100 | 2 | I am working with the definition of a probability generator of a Feller process as stated in Liggett's book, "Continuous time Markov processes":
Let $S$ be a compact state space and denote by $C(S)$ the continuous functions on $S$. A probability generator is a linear operator $L$ satisfying:
i) $\mathcal{D}(L)$ is ... | https://mathoverflow.net/users/43839 | Intuition for the definition of a probability generator of a Feller process | Property ii) is natural to ask if you keep in mind the martingale characterization of the generator. Indeed, let $x$ such that $f(x)=\inf f$. Let now $X\_t$ be a Markov process with generator $L$ started at $x$. We have
$\mathbb{E}(f(X\_t) )\ge f(x)$.
But the process $f(X\_t) -\int\_0^t Lf(X\_s) ds$ is a martingal... | 4 | https://mathoverflow.net/users/48356 | 176103 | 88,756 |
https://mathoverflow.net/questions/176078 | 14 | Let $\mathbf{Set}$ be the category of finite sets and functions between them, and let $\mathbf{Vect}$ be the category of finite-dimensional complex vector spaces and linear transformations between them. There is a free functor $G \colon \mathbf{Set} \to \mathbf{Vect}$, left adjoint to the forgetful functor, given by $G... | https://mathoverflow.net/users/10368 | Are all vector-space valued functors on sets free? | This is probably an absurdly over-complicated answer, but ...
Let
$$J(X)=\left\{\sum\_{x\in X}a\_xx\in GX: \sum\_{x\in X}a\_x=0\right\}.$$
I claim that $J$ is not of the form $H\circ G\circ F$.
Suppose it were.
Let $n=\{0,\dots,n-1\}$.
The functors $F,G,H$ induce group homomorphisms
$$S(n)\to S(F(n))\to \op... | 18 | https://mathoverflow.net/users/22989 | 176109 | 88,759 |
https://mathoverflow.net/questions/176108 | 4 | here is my question:
I work over the field of complex numbers, and my schemes are separated and of finite type. Let $X$ be a quasi projective scheme (but not projective), let $Y$ be a regular projective scheme, and let $f: X \to Y$ be a monomorphism of schemes (but not necessary an immersion).
Is the pull-back by ... | https://mathoverflow.net/users/23236 | Pull-Back of an ample line bundle | A monomorphism is injective (EGA IV 17.2.6), hence quasi-finite, hence quasi-affine (Zariski main theorem). Now the pull-back of an ample sheaf by a quasi-affine morphism is ample (EGA II 5.1.12).
| 8 | https://mathoverflow.net/users/40297 | 176115 | 88,760 |
https://mathoverflow.net/questions/176116 | 7 | Little is known on general conditions guaranteeing that the exponential map between a Lie algebra and an associated Lie group is surjective.
Let $M$ be a noncompact connected Riemann manifold, and $G$ be its (Lie) group of isometries. Does anybody know whether the exponential map between the Lie algebra of $G$ and th... | https://mathoverflow.net/users/54780 | The surjectivity of the exponential map for the isometry group | No. Consider, for example, $M=\mathrm{SL}(3,\mathbb{R})/\mathrm{SO(3)}$ endowed with its $\mathrm{SL}(3,\mathbb{R})$-invariant Riemannian metric $g$ (which is unique up to a positive constant multiple). This is an irreducible symmetric space of noncompact type. The identity component of the isometry group of $(M,g)$ is... | 16 | https://mathoverflow.net/users/13972 | 176120 | 88,761 |
https://mathoverflow.net/questions/175958 | 4 | What is the internal language of the category Cat of small categories?
I found an [article](http://www.brics.dk/RS/01/27/) by Glynn Winskel and his student Mario Jose Cáccamo about such calculus! However it is limited to a fragment of Cat. I wonder if there have been further work since the PhD thesis of Cáccamo.
<h... | https://mathoverflow.net/users/55915 | internal language for the 2-category of small categories | You may be interested in Harper and Licata, *2-Dimensional Directed Dependent Type Theory*, [PDF](http://dlicata.web.wesleyan.edu/pubs/lh102dtt/lh102dtt.pdf). I did some abortive work on this [myself](http://ncatlab.org/michaelshulman/show/internal+logic+of+a+2-category) as well.
| 6 | https://mathoverflow.net/users/49 | 176124 | 88,762 |
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